On the Optimal Designof Participating Life Insurance Contracts

Pietro MillossovichCass Business School, City, University of London

and DEAMS, University of Trieste(joint with Chiara Corsato and Anna Rita Bacinello)

OICA Conference 2020

1

Aim

• Investigate how policyholders and shareholders contribute to theformation of a life insurance company

• Focus on stylized, participating contracts

• policyholders max their preferences over

B contribution rateB minimum guaranteed

for given participation rate

• role of financial and (systematic) longevity risk ⇒ focus on a largeportfolio

• discuss insurance demand - role of regulatory environment (SII)constraints

B fair pricingB solvency

2

Aim

• Investigate how policyholders and shareholders contribute to theformation of a life insurance company

• Focus on stylized, participating contracts

• policyholders max their preferences over

B contribution rateB minimum guaranteed

for given participation rate

• role of financial and (systematic) longevity risk ⇒ focus on a largeportfolio

• discuss insurance demand - role of regulatory environment (SII)constraints

B fair pricingB solvency

2

Aim

• Investigate how policyholders and shareholders contribute to theformation of a life insurance company

• Focus on stylized, participating contracts

• policyholders max their preferences over

B contribution rateB minimum guaranteed

for given participation rate

• role of financial and (systematic) longevity risk ⇒ focus on a largeportfolio

• discuss insurance demand - role of regulatory environment (SII)constraints

B fair pricingB solvency

2

References - Optimal Insurance

• Non Life Insurance (. . .)

• (Participating) Life insurance

B Briys and de Varenne, GPRIT (1994), JRI (1997) ⇒ extended in manydirections by Jorgensen, Le Courtois, Quittard-Pinon, A. Chen,Bernard, . . .⇒ stochastic interest rates, continuous regulatorymonitoring, surrender, . . .

B Bacinello et al., EAJ (2018)B Schmeiser and Wagner, JRI (2015), Braun et al. JRF (2019)B Chen and Hieber, Astin (2016)B Gatzert et al., JRI (2012)B Huang et al., JRI (2008)

3

References - Optimal Insurance

• Non Life Insurance (. . .)

• (Participating) Life insurance

B Briys and de Varenne, GPRIT (1994), JRI (1997) ⇒ extended in manydirections by Jorgensen, Le Courtois, Quittard-Pinon, A. Chen,Bernard, . . .⇒ stochastic interest rates, continuous regulatorymonitoring, surrender, . . .

B Bacinello et al., EAJ (2018)B Schmeiser and Wagner, JRI (2015), Braun et al. JRF (2019)B Chen and Hieber, Astin (2016)B Gatzert et al., JRI (2012)B Huang et al., JRI (2008)

3

Contract features - finite portfolio

• Insurer’s capital structure at t = 0

Assets LiabilitiesW0 L0 = αW0 = global premium

E0 = (1− α)W0 = equityholders’ contribution

W0 W0

• Pure endowment with maturity T

• 0 ≤ α ≤ 1: leverage ratio

• G ≥ 0: guaranteed amount per £ insured

• 0 ≤ δ ≤ 1: participation rate

4

Contract features - finite portfolio

• Insurer’s capital structure at t = 0

Assets LiabilitiesW0 L0 = αW0 = global premium

E0 = (1− α)W0 = equityholders’ contribution

W0 W0

• Pure endowment with maturity T

• 0 ≤ α ≤ 1: leverage ratio

• G ≥ 0: guaranteed amount per £ insured

• 0 ≤ δ ≤ 1: participation rate

4

Contract features - finite portfolio

• N0 initial (homogeneous) policyholders

• N =∑N0

i=1 1Ei survivors at T (Ei = ‘policyholder i is alive at T ’)

• WT = W0eR, assets at maturity, R: log-return

• Guaranteed global payment

GT =αW0

N0GN

• Global liability at T : following (Briys and de Varenne, 1994, 1997),on {N > 0}

LT = GT︸︷︷︸guaranteed

+ δ(αWT −GT

)+

︸ ︷︷ ︸bonus

−(GT −WT

)+

︸ ︷︷ ︸default

5

Contract features - finite portfolio

• N0 initial (homogeneous) policyholders

• N =∑N0

i=1 1Ei survivors at T (Ei = ‘policyholder i is alive at T ’)

• WT = W0eR, assets at maturity, R: log-return

• Guaranteed global payment

GT =αW0

N0GN

• Global liability at T : following (Briys and de Varenne, 1994, 1997),on {N > 0}

LT = GT︸︷︷︸guaranteed

+ δ(αWT −GT

)+

︸ ︷︷ ︸bonus

−(GT −WT

)+

︸ ︷︷ ︸default

5

Contract features - finite portfolio

• N0 initial (homogeneous) policyholders

• N =∑N0

i=1 1Ei survivors at T (Ei = ‘policyholder i is alive at T ’)

• WT = W0eR, assets at maturity, R: log-return

• Guaranteed global payment

GT =αW0

N0GN

• Global liability at T : following (Briys and de Varenne, 1994, 1997),on {N > 0}

LT = GT︸︷︷︸guaranteed

+ δ(αWT −GT

)+

︸ ︷︷ ︸bonus

−(GT −WT

)+

︸ ︷︷ ︸default

5

Pricing assumptions

• P historical measure

B R has a distribution with support RB Conditionally on 0 < Π < 1 the Ei’s are i.i.d. with

P(Ei|Π) = Π

B R independent of Π, (Ei)

• P ∼ P, insurer’s pricing measure

B Risk neutrality condition: E[eR] = erT , r = risk-free interest rate

B Conditionally on 0 < Π < 1 the Ei’s are i.i.d. with

P(Ei|Π) = Π

B R independent of Π, (Ei)

• Π, Π: (systematic) longevity risk

6

Pricing assumptions

• P historical measure

B R has a distribution with support RB Conditionally on 0 < Π < 1 the Ei’s are i.i.d. with

P(Ei|Π) = Π

B R independent of Π, (Ei)

• P ∼ P, insurer’s pricing measure

B Risk neutrality condition: E[eR] = erT , r = risk-free interest rate

B Conditionally on 0 < Π < 1 the Ei’s are i.i.d. with

P(Ei|Π) = Π

B R independent of Π, (Ei)

• Π, Π: (systematic) longevity risk

6

Pricing assumptions

• P historical measure

B R has a distribution with support RB Conditionally on 0 < Π < 1 the Ei’s are i.i.d. with

P(Ei|Π) = Π

B R independent of Π, (Ei)

• P ∼ P, insurer’s pricing measure

B Risk neutrality condition: E[eR] = erT , r = risk-free interest rate

B Conditionally on 0 < Π < 1 the Ei’s are i.i.d. with

P(Ei|Π) = Π

B R independent of Π, (Ei)

• Π, Π: (systematic) longevity risk

6

Pricing measure P vs historical measure P

• risk premium for financial risk

P(R) �st P(R)

⇒ E[eR] ≥ erT

• loading for (systematic) longevity risk

P(Π) �st P(Π)

⇒ P(Ei) ≥ P(Ei)

7

Pricing measure P vs historical measure P

• risk premium for financial risk

P(R) �st P(R)

⇒ E[eR] ≥ erT

• loading for (systematic) longevity risk

P(Π) �st P(Π)

⇒ P(Ei) ≥ P(Ei)

7

Large portfolio

• Global liability in a finite portfolio: on {N > 0}

LT = GT︸︷︷︸guaranteed

+ δ(αWT −GT

)+

︸ ︷︷ ︸bonus

−(GT −WT

)+

︸ ︷︷ ︸default

• Assumption: as N0 → +∞

W0

N0→ w0(> 0)

w0 = assets per individual contract in a large portfolio

• LLN for exchangeable sequences: as N0 → +∞

N

N0→ Π a.s. under P (→ Π a.s. under P)

8

Large portfolio

• Global liability in a finite portfolio: on {N > 0}

LT = GT︸︷︷︸guaranteed

+ δ(αWT −GT

)+

︸ ︷︷ ︸bonus

−(GT −WT

)+

︸ ︷︷ ︸default

• Assumption: as N0 → +∞

W0

N0→ w0(> 0)

w0 = assets per individual contract in a large portfolio

• LLN for exchangeable sequences: as N0 → +∞

N

N0→ Π a.s. under P (→ Π a.s. under P)

8

Large portfolio

• Global liability in a finite portfolio: on {N > 0}

LT = GT︸︷︷︸guaranteed

+ δ(αWT −GT

)+

︸ ︷︷ ︸bonus

−(GT −WT

)+

︸ ︷︷ ︸default

• Assumption: as N0 → +∞

W0

N0→ w0(> 0)

w0 = assets per individual contract in a large portfolio

• LLN for exchangeable sequences: as N0 → +∞

N

N0→ Π a.s. under P (→ Π a.s. under P)

8

Large portfolio

• Individual liability in a large portfolio: on Ei

`(i) = limN0→+∞

LTN

= αw0G︸ ︷︷ ︸guaranteed

+ δαw0

(eR

Π−G

)+

︸ ︷︷ ︸bonus

−w0

(αG− eR

Π

)+

︸ ︷︷ ︸default

a.s. under P

• Same holds under P with Π replaced by Π

9

Large portfolio

• Individual liability in a large portfolio: on Ei

`(i) = limN0→+∞

LTN

= αw0G︸ ︷︷ ︸guaranteed

+ δαw0

(eR

Π−G

)+

︸ ︷︷ ︸bonus

−w0

(αG− eR

Π

)+

︸ ︷︷ ︸default

a.s. under P

• Same holds under P with Π replaced by Π

9

Individual liability profile

10

The Policyholders’ Problem

• Policyholders’ preferences

u : VNM utility function, u′ > 0, u′′ < 0

• (representative) Policyholder’s decision problem

maxα,G

E[u(

e−rT `(i) − αw0︸ ︷︷ ︸NPV

)]

• Regulatory constraintsB maximum guaranteedB fair pricingB solvency based capital allocation criterion

11

The Policyholders’ Problem

• Policyholders’ preferences

u : VNM utility function, u′ > 0, u′′ < 0

• (representative) Policyholder’s decision problem

maxα,G

E[u(

e−rT `(i) − αw0︸ ︷︷ ︸NPV

)]

• Regulatory constraintsB maximum guaranteedB fair pricingB solvency based capital allocation criterion

11

The Policyholders’ Problem

• Policyholders’ preferences

u : VNM utility function, u′ > 0, u′′ < 0

• (representative) Policyholder’s decision problem

maxα,G

E[u(

e−rT `(i) − αw0︸ ︷︷ ︸NPV

)]

• Regulatory constraintsB maximum guaranteedB fair pricingB solvency based capital allocation criterion

11

The Policyholders’ Problem

• Regulatory constraintsB Regulatory cap on minimum guaranteed

G ≤ G

B Fairness conditionαw0 = E

[e−rT `(i)

]B Solvency constraint, 0 < ε < 1

P(w0eR < w0αGΠ

)≤ ε

12

The Policyholders’ Problem

• Regulatory constraintsB Regulatory cap on minimum guaranteed

G ≤ G

B Fairness conditionαw0 = E

[e−rT `(i)

]

B Solvency constraint, 0 < ε < 1

P(w0eR < w0αGΠ

)≤ ε

12

The Policyholders’ Problem

• Regulatory constraintsB Regulatory cap on minimum guaranteed

G ≤ G

B Fairness conditionαw0 = E

[e−rT `(i)

]B Solvency constraint, 0 < ε < 1

P(w0eR < w0αGΠ

)≤ ε

12

The Policyholders’ Problem

• Policyholder’s decision problem

maxα,G

E[u(e−rT `(i) − αw0

)]

• Solution (α∗, G∗)

• can be seen as optimal allocation problem

• optimal contracts include Pareto efficient contracts (criteria: expectedutility for ph, ruin prob for insurer)

13

The Policyholders’ Problem

• Policyholder’s decision problem

maxα,G

E[u(e−rT `(i) − αw0

)]

• Solution (α∗, G∗)

• can be seen as optimal allocation problem

• optimal contracts include Pareto efficient contracts (criteria: expectedutility for ph, ruin prob for insurer)

13

Regulatory constraints (δ < 1)

α

G

g(0)=G

g(α)

0 1

αG=xε

G

erT

E~(Π~)

14

Preliminaries

• Which factors drive the insurance demand: α∗ = 0 or α∗ > 0?

• Distinguish partial participation δ < 1 from full participation δ = 1

Lemma

if P(R) = P(R) then α∗ = 0,

hence assumeP(R) ≺st P(R)

15

Preliminaries

• Which factors drive the insurance demand: α∗ = 0 or α∗ > 0?

• Distinguish partial participation δ < 1 from full participation δ = 1

Lemma

if P(R) = P(R) then α∗ = 0,

hence assumeP(R) ≺st P(R)

15

Full Participation (δ = 1)

• Fairness: α = 0 or G = 0 or α = 1

Theorem

α∗ > 0 alwaysα∗ = 1 (mutual company) iff

E[u′(w0(J − 1))(J − 1)Π− u′(−w0)(1−Π)

]≥ 0,

J = eR−rT

Π

• E.g., CARA, α∗ < 1 when risk aversion is small

16

Full Participation (δ = 1)

• Fairness: α = 0 or G = 0 or α = 1

Theorem

α∗ > 0 alwaysα∗ = 1 (mutual company) iff

E[u′(w0(J − 1))(J − 1)Π− u′(−w0)(1−Π)

]≥ 0,

J = eR−rT

Π

• E.g., CARA, α∗ < 1 when risk aversion is small

16

Partial participation (δ < 1)

• Fairness: G = g(α) for α > 0

Theorem: δ = 0 (traditional policy)

When E[Π] < E[Π], there exists G′ ≥ erT /E[Π] st α∗ = 0 for all G ≤ G′

Theorem: 0 < δ < 1

There exists d(G) ↓ G with d(G) > 0 iff G < erT /E[Π] st

1 if 0 < δ ≤ d(G) then α∗ = 0,

2 if δ > d(G) and

g(0)E[Π]

+ δE[(eR − g(0)Π)+

]> erT , (∗)

then α∗ > 0

17

Partial participation (δ < 1)

• Fairness: G = g(α) for α > 0

Theorem: δ = 0 (traditional policy)

When E[Π] < E[Π], there exists G′ ≥ erT /E[Π] st α∗ = 0 for all G ≤ G′

Theorem: 0 < δ < 1

There exists d(G) ↓ G with d(G) > 0 iff G < erT /E[Π] st

1 if 0 < δ ≤ d(G) then α∗ = 0,

2 if δ > d(G) and

g(0)E[Π]

+ δE[(eR − g(0)Π)+

]> erT , (∗)

then α∗ > 0

17

. . . Partial participation (δ < 1)• Fairness: G = g(α) for α > 0• when does the condition hold?

Corollary

1 There exists 0 < δ′ < 1 st α∗ > 0 for all δ > δ′

2 For δ > d(G) and Π “close” to Π, then α∗ > 0

Corollary: binding constraints

1 If for G > 0 and d(G) < δ < 1 condition (∗) holds, there exists ε′ > 0 st forall 0 < ε ≤ ε′,

P(w0eR < w0α∗G∗Π) = ε,

2 If for 0 < δ < 1 condition (∗) holds , there exists G′> g(0) st

G∗ = g(α∗) = G

for all g(0) < G < G′.

18

. . . Partial participation (δ < 1)• Fairness: G = g(α) for α > 0• when does the condition hold?

Corollary

1 There exists 0 < δ′ < 1 st α∗ > 0 for all δ > δ′

2 For δ > d(G) and Π “close” to Π, then α∗ > 0

Corollary: binding constraints

1 If for G > 0 and d(G) < δ < 1 condition (∗) holds, there exists ε′ > 0 st forall 0 < ε ≤ ε′,

P(w0eR < w0α∗G∗Π) = ε,

2 If for 0 < δ < 1 condition (∗) holds , there exists G′> g(0) st

G∗ = g(α∗) = G

for all g(0) < G < G′.

18

. . . Partial participation (δ < 1)• Fairness: G = g(α) for α > 0• when does the condition hold?

Corollary

1 There exists 0 < δ′ < 1 st α∗ > 0 for all δ > δ′

2 For δ > d(G) and Π “close” to Π, then α∗ > 0

Corollary: binding constraints

1 If for G > 0 and d(G) < δ < 1 condition (∗) holds, there exists ε′ > 0 st forall 0 < ε ≤ ε′,

P(w0eR < w0α∗G∗Π) = ε,

2 If for 0 < δ < 1 condition (∗) holds , there exists G′> g(0) st

G∗ = g(α∗) = G

for all g(0) < G < G′.

18

Conclusions & Extensions

• Main features:

B Analytical yet stylized model for participating life officesB General policyholder’s preferencesB effect of regulator (G, solvency, ruin) or corporate (δ, σ) on insurance

demandB effect of (systematic) longevity risk

• Extensions:

B Stochastic interest ratesB Dynamic modelB Deferred (guaranteed) annuities vs pure endowment?B Asymmetric information

19

Conclusions & Extensions

• Main features:

B Analytical yet stylized model for participating life officesB General policyholder’s preferencesB effect of regulator (G, solvency, ruin) or corporate (δ, σ) on insurance

demandB effect of (systematic) longevity risk

• Extensions:

B Stochastic interest ratesB Dynamic modelB Deferred (guaranteed) annuities vs pure endowment?B Asymmetric information

19